APPENDIX 1. THE ATOMIC, NUCLEAR AND BLACK HOLE DENSITIES

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1 7 February ore APPENDIX 1. THE ATOMIC, NUCLEAR AND BLACK HOLE DENSITIES The Schwarzschild formula remains as it is despite all developments [1] in the physics of Black Holes including what has been discovered by RQST. If we put in (1) the masses of the Sun, m, of our Earth, m #$!, of our Moon, m "#, the result is R 3 km R #$! 0,9 cm R "# 0,1 mm. The Schwarzschild formula establishes a relation between gravitational masses and Radius of a Black Hole. If we were able to put the mass of our Sun inside a sphere of radius of about 3 km, the Sun will become a Black Hole. The same holds for the masses of our Earth and of our Moon, if we were able to concentrate these masses, insides spheres of radii, approximately equal to 0,9 cm and 0,1 mm, respectively. 1

2 The reason why none of these bodies will ever became a Black Hole is due to the fact that their masses consist of protons, neutrons and electrons which have to obey the laws established by the existence of the Fundamental Forces which are not Gravitational, i.e. SU(3) SU(2) U(1). If these forces would not to be there, the three masses, m, m #$!, m "# would indeed be Black Holes, if concentrated in the correspondent spheres with the radii given by Schwarzschild formula (1). In a Universe where the Fundamental Forces were only the Einstein Gravitational Forces, the spectrum of all possible masses obeys the Schwarzschild formula (1). This equation has as the smallest limit the Planck conditions. No limit is given for the largest Black Hole. No conditions to this limit are imposed by RQST. The very instant of the Big Bang is the Planck Time and the corresponding density of the Universe is the Planck density. THE ATOMIC DENSITY: ρ atomic In our world the density is dictated by the fact that we are made by atoms and therefore the basic quantities are the mass 2

3 of the nucleon (m N gr) and the radius of the atom (10 8 cm) which is dictated by the electric charge and the mass of the electron. Let us call the value of this density the atomic density : ρ #$%& M #$%&!! R #$ 10! gr 10!! cm! = 10! gr 10! cm! = 1 gr cm!. The density of water is typical of the atomic density. In the above formula we neglect details like [(4/3) π] in front of! R #$ to estimate the atomic volume. When we go from water to lead the atomic density increases by an order of magnitude. This is due to the increase in the mass of the nucleus by two orders of magnitude m nucleus Pb 10 2 M nucleon and a correspondent increase by an order of magnitude in the atomic volume. THE NUCLEAR DENSITY: ρ nuclear The next possible density many orders of magnitudes higher is the nuclear density: times greater than the atomic density. 3

4 The reason being the value of the nuclear radius, which is of the order of one Fermi-unit (10 13 cm), i.e. five orders of magnitude smaller than the atomic radius. Atomic and nuclear bindings are specific for a given Element of the Mendeleev Table and do not change when the amount of the given Element changes. For both forms of matter, atomic and nuclear, the density does not change when the amount of matter increases: one ton of lead has the same density as one kilogram of lead. THE BLACK HOLES DENSITIES: ρ Black Holes For matter where the binding force is gravitational (without any other forces being involved), the density decreases when the amount of mass increases. More precisely the density decreases with the square of the mass. This is the great discovery of Schwarzschild and is coming from the relation (1) which exists between the Radius of a Black Hole, R, and the mass of the same Black Hole, M. 4

5 REFERENCES [1] Quantum Gravity without Space-Time Singularities or Horizons G. t Hooft, in Proceedings Subnuclear Physics Erice School 2009, Vol. 47, World Scientific (2011); Latest News from Black-Holes Physics G. t Hooft, in Proceedings Subnuclear Physics Erice School 2010, Vol. 48, World Scientific (2013); Using Black Holes to Understand Quantum Gravity G. t Hooft, in Proceedings Subnuclear Physics Erice School 2011, Vol. 49, World Scientific (2013); Beyond Relativistic Quantum String Theory G. t Hooft, in Proceedings Subnuclear Physics Erice School 2012, Vol. 50, World Scientific (2014). 5